Heat Equation In Unbounded Domains Research Articles

Averaged controllability of heat equation in unbounded domains with random geometry and location of controls: The Green’s function approach

Averaged controllability of heat equation in unbounded domains with random geometry and location of controls: The Green’s function approach

Distributed controllability of one-dimensional heat equation in unbounded domains: The Green’s function approach

Distributed controllability of one-dimensional heat equation in unbounded domains: The Green’s function approach

Examples of the nonuniqueness of solutions of the mixed problem for the heat equation in unbounded domains

For a wide class of domains of revolution, we construct examples of the nonuniqueness of solutions of the first mixed problem for the heat equation, which supports the exactness of a uniqueness class of Tacklind type.

High-Order Local Absorbing Boundary Conditions for Heat Equation in Unbounded Domains

With the development of numerical methods the numerical computations require higher and higher accuracy. This paper is devoted to the high-order local absorbing boundary conditions (ABCs) for heat equation. We proved that the coupled system yields a stable problem between the obtained high-order local ABCs and the partial differential equation in the computational domain. This method has been used widely in wave propagation models only recently. We extend the spirit of the methodology to parabolic ones, which will become a basis to design the local ABCs for a class of nonlinear PDEs. Some numerical tests show that the new treatment is very efficient and tractable.

Identification of the class of initial data for the insensitizing control of the heat equation

This paper is devoted to analyze the class of initial data that can be insensitized for the heat equation. This issue has been extensively addressed in the literature both in the case of complete and approximate insensitization (see [19] and [1], respectively). But in the context of pure insensitization there are very few results identifying the class of initial data that can be insensitized. This is a delicate issue which is related to the fact thatinsensitization turns out to be equivalent to suitableobservability estimates for a coupled system of heat equations,one being forward and the other one backward in time. The existing Carleman inequalities techniques can be applied but they only give interior information of the solutions, which hardly allows identifying the initial data because of the strong irreversibility of the equations involved in the system, one of them being an obstruction at the initial time $t=0$ and the other one at the final one $t=T$.In this article we consider different geometric configurations in which the subdomains to be insensitized and the one in which the external control acts play a key role. We show that, under rather restrictive geometric restrictions, initial data in aclass that can be characterized in terms of a summability condition of their Fourier coefficients with suitable weights, can be insensitized. But, the main result of the paper, which might seem surprising, shows that this fails to be true in general, so that even the first eigenfunction of the system can not be insensitized. This result is similar to those obtained in the context of the null controllability of the heat equation in unbounded domains in [14] where it is shown that smooth and compactly supported initial data may not be controlled.Our proofs combine the existing observability results for heat equations obtained by means of Carleman inequalities, energy and gaussian estimates and Fourier expansions.

Some results on controllability for linear and nonlinear heat equations in unbounded domains

In this paper we present some results concerning the null controllability for a heat equation in unbounded domains. We characterize the conditions that must satisfy the auxiliary function that leads to a global Carleman inequality for the adjoint problem and then to get a null controllability result. We give some examples of unbounded domains $(\Omega,\omega)$ that satisfy these sufficient conditions. Finally, when $\Omega\setminus \overline \omega$ is bounded, we prove the null controllability of the semilinear heat equation when the nonlinearity $f(y,\nabla y)$ grows more slowly than $|y|\log^{3/2}(1+|y|+|\nabla y|)+ |\nabla y|\log^{1/2}(1+|y|+|\nabla y|)$ at infinity (generally in this case in the absence of control, blow-up occurs). In this aim we also prove the linear null controllability problem with $L^\infty$-controls.

Numerical solution of the heat equation with nonlinear boundary conditions in unbounded domains

AbstractThe numerical solution of the heat equation in unbounded domains (for a 1D problem‐semi‐infinite line and for a 2D one semi‐infinite strip) is considered. The artificial boundaries are introduced and the exact artificial boundary conditions are derived. The original problems are transformed into problems on finite domains. The space semi‐discretization by finite element method and the full approximation by the implicit‐explicit Euler's method are presented. The solvability of the full discretization schemes is analyzed. Computational examples demonstrate the accuracy and the efficiency of the algorithms. Also, the behavior of blowing up solutions is examined numerically. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 379–399, 2007

On the Controllability of a Fractional Order Parabolic Equation

The null-controllability property of a $1-d$ parabolic equation involving a fractional power of the Laplace operator, $(-\Delta)^\alpha$, is studied. The control is a scalar time-dependent function $g=g(t)$ acting on the system through a given space-profile $f=f(x)$ on the interior of the domain. Thus, the control g determines the intensity of the space control f applied to the system, the latter being given a priori. We show that, if $\alpha\leq 1/2$ and the shape function f is, say, in $L^2$, no initial datum belonging to any Sobolev space of negative order may be driven to zero in any time. This is in contrast with the existing positive results for the case $\alpha >1/2$ and, in particular, for the heat equation that corresponds to $\alpha=1$. This negative result exhibits a new phenomenon that does not arise either for finite-dimensional systems or in the context of the heat equation. On the contrary, if more regularity of the shape function f is assumed, then we show that there are initial data in any Sobolev space $H^m$ that may be controlled. Once again this is precisely the opposite behavior with respect to the control properties of the heat equation in which, when increasing the regularity of the control profile, the space of controllable data decreases. These results show that, in order for the control properties of the heat equation to be true, the dynamical system under consideration has to have a sufficiently strong smoothing effect that is critical when $\alpha=1/2$ for the fractional powers of the Dirichlet Laplacian in $1-d$. The results we present here are, in nature and with respect to techniques of proof, similar to those on the control of the heat equation in unbounded domains in [S. Micu and E. Zuazua, Trans. Amer. Math. Soc., 353 (2000), pp. 1635-1659] and [S. Micu and E. Zuazua, Portugal. Math., 58 (2001), pp. 1-24]. We also discuss the hyperbolic counterpart of this problem considering a fractional order wave equation and some other models.

On the null-controllability of the heat equation in unbounded domains

We make two remarks about the null-controllability of the heat equation with Dirichlet condition in unbounded domains. Firstly, we give a geometric necessary condition (for interior null-controllability in the Euclidean setting) which implies that one cannot go infinitely far away from the control region without tending to the boundary (if any), but also applies when the distance to the control region is bounded. The proof builds on heat kernel estimates. Secondly, we describe a class of null-controllable heat equations on unbounded product domains. Elementary examples include an infinite strip in the plane controlled from one boundary and an infinite rod controlled from an internal infinite rod. The proof combines earlier results on compact manifolds with a new lemma saying that the null-controllability of an abstract control system and its null-controllability cost are not changed by taking its tensor product with a system generated by a non-positive self-adjoint operator.

Convergence of difference scheme for heat equation in unbounded domains using artificial boundary conditions

The finite difference solution of one-dimensional heat conduction equation in unbounded domains is considered. An artificial boundary is introduced to make the computational domain finite. On the artificial boundary an exact boundary condition is applied to reduce the original problem to an initial-boundary value problem. A finite difference scheme is constructed by the method of reduction of order. It is proved that the finite difference scheme is uniquely solvable, unconditionally stable and convergent with the order 2 in space and the order 3/2 in time under an energy norm. A numerical example demonstrates the theoretical results.

Null controllability of the heat equation in unbounded domains by a finite measure control region

Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes and Zuazua, we study the null controllability of the heat equation in unbounded domains, typically or . Considering an unbounded and disconnected control region of the form , we prove two null controllability results: under some technical assumption on the control parts , we prove that every initial datum in some weighted L2 space can be controlled to zero by usual control functions, and every initial datum in L2(Ω) can be controlled to zero using control functions in a weighted L2 space. At last we give several examples in which the control region has a finite measure and our null controllability results apply.

Convergence of difference scheme for heat equation in unbounded domains using artificial boundary conditions

The finite difference solution of one-dimensional heat conduction equation in unbounded domains is considered. An artificial boundary is introduced to make the computational domain finite. On the artificial boundary an exact boundary condition is applied to reduce the original problem to an initial-boundary value problem. A finite difference scheme is constructed by the method of reduction of order. It is proved that the finite difference scheme is uniquely solvable, unconditionally stable and convergent with the order 2 in space and the order 3/2 in time under an energy norm. A numerical example demonstrates the theoretical results.

Persistent regional null contrillability for a class of degenerate parabolic equations

Motivated by physical models and the so-called Crocco equation, we study the controllability properties of a class of degenerate parabolic equations. Due to degeneracy, classical null controllability results do not hold for this problem in general.<br> First, we prove that we can drive the solution to rest at time $T$ in a suitable subset of the space domain (regional null controllability). However, unlike for nondegenerate parabolic equations, this property is no more automatically preserved with time. Then, we prove that, given a time interval $(T,T')$, we can control the equation up to $T'$ and remain at rest during all the time interval $(T,T')$ on the same subset of the space domain (persistent regional null controllability). The proofs of these results are obtained via new observability inequalities derived from classical Carleman estimates by an appropriate use of cut-off functions.<br> With the same method, we also derive results of regional controllability for a Crocco type linearized equation and for the nondegenerate heat equation in unbounded domains.

A class of artificial boundary conditions for heat equation in unbounded domains

In this paper, the numerical solutions of three problems of heat equation on unbounded domains are considered. For each problem, we introduce an artificial boundary Γ to make the computational domain finite. On the artificial boundary Γ, we propose an exact artificial boundary condition to reduce the original problem to an initial-boundary problem of heat equation on the finite computational domain, which is equivalent to the original problem. Then the finite difference method and finite element method are used to solve the reduced problem on the finite computational domain. In the end of this paper, three numerical examples show the feasibility and effectiveness of the method given in this paper.

On a Boundary Integral Method for the Solution of the Heat Equation in Unbounded Domains with a Nonsmooth Boundary

On a Boundary Integral Method for the Solution of the Heat Equation in Unbounded Domains with a Nonsmooth Boundary

Spectral Approximation of the Free-Space Heat Kernel

Many problems in applied mathematics, physics, and engineering require the solution of the heat equation in unbounded domains. Integral equation methods are particularly appropriate in this setting for several reasons: they are unconditionally stable, they are insensitive to the complexity of the geometry, and they do not require the artificial truncation of the computational domain as do finite difference and finite element techniques. Methods of this type, however, have not become widespread due to the high cost of evaluating heat potentials. When mpoints are used in the discretization of the initial data, Mpoints are used in the discretization of the boundary, and Ntime steps are computed, an amount of work of the order O(N2M2+NMm) has traditionally been required. In this paper, we present an algorithm which requires an amount of work of the order O(NMlog M+mlog m) and which is based on the evolution of the continuousspectrum of the solution. The method generalizes an earlier technique developed by Greengard and Strain (1990, Comm. Pure Appl. Math.43, 949) for evaluating layer potentials in bounded domains.

Approximate controllability of a semilinear heat equation in unbounded domains

Approximate controllability of a semilinear heat equation in unbounded domains

Controls insensitizing the norm of the solution of a semilinear heat equation in unbounded domains

We consider a semilinear heat equation in an unbounded domain Ω with partially known initial data. The insensitizing problem consists in finding a control function such that some functional of the state is locally insensitive to the perturbations of these initial data. For bounded domains Bodart and Fabre proved the existence of insensitizing controls of the norm of the observation of the solution in an open subset of the domain. In this paper we prove similar results when Ω is unbounded. We consider the problem in bounded domains of the form Ωr = Ω ∩ Br where Br denotes the ball centered in zero of radius r We show that for r large enough the control proposed by Bodart and Fabre for the problem in Ωr, provides an insensitizing control for our problem in Ω.